Example of Straight-Forward Approach in Algebra

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What is the relationship between the three statements r, s, and t in the proof of the theorem?

They are mutually exclusive

They are collectively exhaustive (correct)

They are equivalent statements

They are independent of each other

What is the value of CD in the case where ∠C is obtuse?

b cos(180° - C) (correct)

b cos C

b cos(90° - C)

What is the expression for a in the case where ∠C is a right angle?

c cos B (correct)

b cos C

b cos C + c cos B

b cos C - c cos B

What is the implication of case (i) on the proof of the theorem?

r ⇒ q is proved Signup and view all the answers

What is the value of BD in the case where ∠C is acute?

c cos B Signup and view all the answers

What is the conclusion that can be drawn from the three cases proved in the theorem?

a = b cos C + c cos B Signup and view all the answers

What is the primary purpose of the straight-forward approach in mathematics?

To establish a connection between a given statement and a conclusion Signup and view all the answers

In the given example, what is the reason behind replacing x2 – 5x + 6 with (x – 3)(x – 2)?

To replace an expression with an equivalent expression Signup and view all the answers

What is the underlying principle in the step 'x – 3 + 3 = 0 + 3 or x – 2 + 2 = 0 + 2'?

The property of adding equal quantities on either side of the equation Signup and view all the answers

What is the conclusion that can be deduced from the statement 'x2 – 5x + 6 = 0'?

x = 3 or x = 2 Signup and view all the answers

What is the role of axioms in the straight-forward approach?

To help in deducing the conclusion from the given statement Signup and view all the answers

What is the advantage of using the straight-forward approach in mathematical proofs?

It provides a logical and systematic approach to mathematical proofs Signup and view all the answers

What is the primary method used to prove the first statement in the text?

Factorization Signup and view all the answers

What is the definition of a one-one function?

A function that maps one element to one element Signup and view all the answers

What is the purpose of the axiom in mathematical induction?

To prove that a proposition is true for all natural numbers Signup and view all the answers

What is the symbolic equivalent of the argumentation process?

p ⇒ q Signup and view all the answers

What is the primary step in proving that the function f(x) = 2x + 5 is one-one?

Assuming that f(x1) = f(x2) and then proving that x1 = x2 Signup and view all the answers

What is the result of dividing both sides of the equation 2x1 + 5 = 2x2 + 5 by 2?

x1 = x2 Signup and view all the answers

What is the symbolic form of the statement 'if a matrix A is invertible, then A is non-singular'?

p ⇒ q Signup and view all the answers

If a matrix A is singular, what is the value of |A|?

0 Signup and view all the answers

What is the contrapositive statement of 'if a matrix A is invertible, then A is non-singular'?

If A is singular, then A is not invertible Signup and view all the answers

What is the purpose of a counter example in proof by a counter example?

To disprove the statement Signup and view all the answers

What is the mathematical notation used to represent the statement 'For each n, 2^(2n) + 1 is a prime'?

∀n ∈ N, 2^(2n) + 1 is a prime Signup and view all the answers

What is the significance of the statement '~ (p ⇒ q)' in the context of proof by a counter example?

It is a disproof of the statement p ⇒ q Signup and view all the answers

What is the conclusion of the proof that the set of all prime numbers is infinite?

The set of all prime numbers is infinite Signup and view all the answers

What is the method of proof used in the first proof?

Proof by contradiction Signup and view all the answers

What is the statement proved in Example 6?

The function f(x) = 2x + 5 is one-one Signup and view all the answers

What is the contrapositive of the statement 'if f(x1) = f(x2), then x1 = x2'?

if x1 ≠ x2, then f(x1) ≠ f(x2) Signup and view all the answers

What is the purpose of the proof by contrapositive?

To prove a statement by proving its equivalent contrapositive Signup and view all the answers

What is the advantage of using proof by contrapositive?

It provides an alternative method to prove a statement Signup and view all the answers